LMFDB - Embedded newform 2160.2.q.l.1441.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(721,2160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.856615824.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 $ x^8 + 11x^6 + 36x^4 + 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1441.1
Root			$-2.06288i$ of defining polynomial
Character	$\chi$	$=$	2160.1441
Dual form			2160.2.q.l.721.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(-2.28651 - 3.96035i) q^{7} +(2.29604 + 3.97686i) q^{11} +(1.83963 - 3.18633i) q^{13} +2.55395 q^{17} -4.76643 q^{19} +(1.28651 - 2.22830i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(0.956412 + 1.65655i) q^{29} +(1.73339 - 3.00231i) q^{31} -4.57301 q^{35} +5.46677 q^{37} +(3.32056 - 5.75139i) q^{41} +(-3.27698 - 5.67589i) q^{43} +(-5.01989 - 8.69471i) q^{47} +(-6.95623 + 12.0485i) q^{49} -4.03813 q^{53} +4.59208 q^{55} +(-6.13567 + 10.6273i) q^{59} +(-4.79604 - 8.30698i) q^{61} +(-1.83963 - 3.18633i) q^{65} +(-5.66972 + 9.82025i) q^{67} -1.67925 q^{71} +3.12530 q^{73} +(10.4998 - 18.1862i) q^{77} +(-6.64113 - 11.5028i) q^{79} +(-5.39275 - 9.34051i) q^{83} +(1.27698 - 2.21179i) q^{85} -4.04905 q^{89} -16.8253 q^{91} +(-2.38322 + 4.12785i) q^{95} +(-8.68962 - 15.0509i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} - q^{7} - q^{11} - 4 q^{13} - 10 q^{17} - 2 q^{19} - 7 q^{23} - 4 q^{25} + 7 q^{29} - 2 q^{31} - 2 q^{35} + 12 q^{37} + 12 q^{41} - 11 q^{43} - 7 q^{47} - 3 q^{49} - 24 q^{53} - 2 q^{55}+ \cdots - q^{97}+O(q^{100}) $ 8 * q + 4 * q^5 - q^7 - q^11 - 4 * q^13 - 10 * q^17 - 2 * q^19 - 7 * q^23 - 4 * q^25 + 7 * q^29 - 2 * q^31 - 2 * q^35 + 12 * q^37 + 12 * q^41 - 11 * q^43 - 7 * q^47 - 3 * q^49 - 24 * q^53 - 2 * q^55 - 11 * q^59 - 19 * q^61 + 4 * q^65 - 10 * q^67 + 24 * q^71 + 18 * q^73 + 32 * q^77 - 24 * q^79 - 23 * q^83 - 5 * q^85 - 42 * q^89 - 28 * q^91 - q^95 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−2.28651	−	3.96035i	−0.864218	−	1.49687i	−0.867821	−	0.496877i	$-0.834480\pi$
$7$	−2.28651	−	3.96035i	−0.864218	−	1.49687i	0.00360263	−	0.999994i	$-0.498853\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.29604	+	3.97686i	0.692282	+	1.19907i	0.971088	+	0.238720i	$0.0767278\pi$
$11$	2.29604	+	3.97686i	0.692282	+	1.19907i	−0.278807	+	0.960347i	$0.589939\pi$
$12$		0			0
$13$	1.83963	−	3.18633i	0.510221	−	0.883728i	−0.489709	−	0.871886i	$-0.662897\pi$
$13$	1.83963	−	3.18633i	0.510221	−	0.883728i	0.999930	−	0.0118424i	$-0.00376965\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.55395			0.619424			0.309712	−	0.950830i	$-0.399767\pi$
$17$	2.55395			0.619424			0.309712	+	0.950830i	$0.399767\pi$
$18$		0			0
$19$	−4.76643			−1.09349			−0.546747	−	0.837298i	$-0.684134\pi$
$19$	−4.76643			−1.09349			−0.546747	+	0.837298i	$0.684134\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.28651	−	2.22830i	0.268255	−	0.464632i	−0.700156	−	0.713990i	$-0.746886\pi$
$23$	1.28651	−	2.22830i	0.268255	−	0.464632i	0.968411	+	0.249358i	$0.0802196\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.956412	+	1.65655i	0.177601	+	0.307614i	0.941058	−	0.338244i	$-0.109833\pi$
$29$	0.956412	+	1.65655i	0.177601	+	0.307614i	−0.763457	+	0.645858i	$0.776500\pi$
$30$		0			0
$31$	1.73339	−	3.00231i	0.311326	−	0.539232i	−0.667324	−	0.744767i	$-0.732560\pi$
$31$	1.73339	−	3.00231i	0.311326	−	0.539232i	0.978650	+	0.205536i	$0.0658937\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.57301			−0.772980
$36$		0			0
$37$	5.46677			0.898732			0.449366	−	0.893348i	$-0.351650\pi$
$37$	5.46677			0.898732			0.449366	+	0.893348i	$0.351650\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.32056	−	5.75139i	0.518585	−	0.898215i	−0.481182	−	0.876621i	$-0.659792\pi$
$41$	3.32056	−	5.75139i	0.518585	−	0.898215i	0.999767	−	0.0215947i	$-0.00687434\pi$
$42$		0			0
$43$	−3.27698	−	5.67589i	−0.499734	−	0.865565i	0.500266	−	0.865872i	$-0.333236\pi$
$43$	−3.27698	−	5.67589i	−0.499734	−	0.865565i	−1.00000		0.000307033i	$0.999902\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.01989	−	8.69471i	−0.732227	−	1.26825i	−0.955929	−	0.293597i	$-0.905148\pi$
$47$	−5.01989	−	8.69471i	−0.732227	−	1.26825i	0.223703	−	0.974657i	$-0.428186\pi$
$48$		0			0
$49$	−6.95623	+	12.0485i	−0.993747	+	1.72122i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.03813			−0.554679			−0.277340	−	0.960772i	$-0.589453\pi$
$53$	−4.03813			−0.554679			−0.277340	+	0.960772i	$0.589453\pi$
$54$		0			0
$55$	4.59208			0.619196
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.13567	+	10.6273i	−0.798796	+	1.38355i	0.121605	+	0.992579i	$0.461196\pi$
$59$	−6.13567	+	10.6273i	−0.798796	+	1.38355i	−0.920401	+	0.390976i	$0.872137\pi$
$60$		0			0
$61$	−4.79604	−	8.30698i	−0.614070	−	1.06360i	−0.990547	−	0.137174i	$-0.956198\pi$
$61$	−4.79604	−	8.30698i	−0.614070	−	1.06360i	0.376477	−	0.926426i	$-0.377135\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.83963	−	3.18633i	−0.228178	−	0.395215i
$66$		0			0
$67$	−5.66972	+	9.82025i	−0.692667	+	1.19973i	0.278294	+	0.960496i	$0.410231\pi$
$67$	−5.66972	+	9.82025i	−0.692667	+	1.19973i	−0.970961	+	0.239238i	$0.923102\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.67925			−0.199291			−0.0996454	−	0.995023i	$-0.531771\pi$
$71$	−1.67925			−0.199291			−0.0996454	+	0.995023i	$0.531771\pi$
$72$		0			0
$73$	3.12530			0.365789			0.182895	−	0.983133i	$-0.441453\pi$
$73$	3.12530			0.365789			0.182895	+	0.983133i	$0.441453\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.4998	−	18.1862i	1.19657	−	2.07251i
$78$		0			0
$79$	−6.64113	−	11.5028i	−0.747185	−	1.29416i	−0.949167	−	0.314773i	$-0.898071\pi$
$79$	−6.64113	−	11.5028i	−0.747185	−	1.29416i	0.201982	−	0.979389i	$-0.435262\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.39275	−	9.34051i	−0.591931	−	1.02525i	−0.993972	−	0.109633i	$-0.965032\pi$
$83$	−5.39275	−	9.34051i	−0.591931	−	1.02525i	0.402041	−	0.915622i	$-0.368301\pi$
$84$		0			0
$85$	1.27698	−	2.21179i	0.138507	−	0.239902i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.04905			−0.429198			−0.214599	−	0.976702i	$-0.568845\pi$
$89$	−4.04905			−0.429198			−0.214599	+	0.976702i	$0.568845\pi$
$90$		0			0
$91$	−16.8253			−1.76377
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.38322	+	4.12785i	−0.244513	+	0.423509i
$96$		0			0
$97$	−8.68962	−	15.0509i	−0.882297	−	1.52818i	−0.848781	−	0.528745i	$-0.822663\pi$
$97$	−8.68962	−	15.0509i	−0.882297	−	1.52818i	−0.0335162	−	0.999438i	$-0.510671\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.57301	+	11.3848i	0.654039	+	1.13283i	0.982134	+	0.188184i	$0.0602603\pi$
$101$	6.57301	+	11.3848i	0.654039	+	1.13283i	−0.328094	+	0.944645i	$0.606406\pi$
$102$		0			0
$103$	−5.75245	+	9.96354i	−0.566806	+	0.981736i	0.430073	+	0.902794i	$0.358488\pi$
$103$	−5.75245	+	9.96354i	−0.566806	+	0.981736i	−0.996879	+	0.0789425i	$0.974846\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.52396			0.534022			0.267011	−	0.963694i	$-0.413964\pi$
$107$	5.52396			0.534022			0.267011	+	0.963694i	$0.413964\pi$
$108$		0			0
$109$	−16.7381			−1.60322			−0.801610	−	0.597847i	$-0.796023\pi$
$109$	−16.7381			−1.60322			−0.801610	+	0.597847i	$0.796023\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.89376	−	3.28009i	0.178150	−	0.308565i	−0.763097	−	0.646284i	$-0.776322\pi$
$113$	1.89376	−	3.28009i	0.178150	−	0.308565i	0.941247	+	0.337719i	$0.109655\pi$
$114$		0			0
$115$	−1.28651	−	2.22830i	−0.119967	−	0.207790i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.83963	−	10.1145i	−0.535318	−	0.927198i
$120$		0			0
$121$	−5.04359	+	8.73575i	−0.458508	+	0.794159i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	11.1444			0.988903			0.494451	−	0.869205i	$-0.335369\pi$
$127$	11.1444			0.988903			0.494451	+	0.869205i	$0.335369\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.9125	−	18.9009i	0.953426	−	1.65138i	0.215496	−	0.976505i	$-0.430863\pi$
$131$	10.9125	−	18.9009i	0.953426	−	1.65138i	0.737930	−	0.674878i	$-0.235804\pi$
$132$		0			0
$133$	10.8985	+	18.8767i	0.945018	+	1.63682i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.22284	−	3.85008i	−0.189910	−	0.328934i	0.755310	−	0.655368i	$-0.227486\pi$
$137$	−2.22284	−	3.85008i	−0.189910	−	0.328934i	−0.945220	+	0.326434i	$0.894153\pi$
$138$		0			0
$139$	5.58338	−	9.67069i	0.473576	−	0.820257i	−0.525967	−	0.850505i	$-0.676296\pi$
$139$	5.58338	−	9.67069i	0.473576	−	0.820257i	0.999542	+	0.0302478i	$0.00962964\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.8954			1.41287
$144$		0			0
$145$	1.91282			0.158851
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.54341	+	11.3335i	−0.536057	+	0.928478i	0.463055	+	0.886330i	$0.346753\pi$
$149$	−6.54341	+	11.3335i	−0.536057	+	0.928478i	−0.999111	+	0.0421478i	$0.986580\pi$
$150$		0			0
$151$	−0.179436	−	0.310792i	−0.0146023	−	0.0252919i	0.858632	−	0.512593i	$-0.171315\pi$
$151$	−0.179436	−	0.310792i	−0.0146023	−	0.0252919i	−0.873234	+	0.487301i	$0.837982\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.73339	−	3.00231i	−0.139229	−	0.241152i
$156$		0			0
$157$	6.53489	−	11.3188i	0.521541	−	0.903335i	−0.478145	−	0.878281i	$-0.658691\pi$
$157$	6.53489	−	11.3188i	0.521541	−	0.903335i	0.999686	−	0.0250544i	$-0.00797591\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.7664			−0.927325
$162$		0			0
$163$	18.7872			1.47152			0.735762	−	0.677240i	$-0.236824\pi$
$163$	18.7872			1.47152			0.735762	+	0.677240i	$0.236824\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.12613	+	12.3428i	−0.551437	+	0.955117i	0.446734	+	0.894667i	$0.352587\pi$
$167$	−7.12613	+	12.3428i	−0.551437	+	0.955117i	−0.998171	+	0.0604500i	$0.980746\pi$
$168$		0			0
$169$	−0.268456	−	0.464980i	−0.0206505	−	0.0357677i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.59208	−	7.95371i	−0.349129	−	0.604710i	0.636966	−	0.770892i	$-0.280189\pi$
$173$	−4.59208	−	7.95371i	−0.349129	−	0.604710i	−0.986095	+	0.166183i	$0.946856\pi$
$174$		0			0
$175$	−2.28651	+	3.96035i	−0.172844	+	0.299374i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18.7872			1.40422			0.702109	−	0.712069i	$-0.252242\pi$
$179$	18.7872			1.40422			0.702109	+	0.712069i	$0.252242\pi$
$180$		0			0
$181$	11.9128			0.885473			0.442737	−	0.896652i	$-0.354008\pi$
$181$	11.9128			0.885473			0.442737	+	0.896652i	$0.354008\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.73339	−	4.73437i	0.200963	−	0.348077i
$186$		0			0
$187$	5.86397	+	10.1567i	0.428816	+	0.742731i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.37285	−	4.10990i	−0.171694	−	0.297382i	0.767318	−	0.641266i	$-0.221591\pi$
$191$	−2.37285	−	4.10990i	−0.171694	−	0.297382i	−0.939012	+	0.343884i	$0.888257\pi$
$192$		0			0
$193$	−4.27698	+	7.40794i	−0.307863	+	0.533235i	−0.977895	−	0.209098i	$-0.932947\pi$
$193$	−4.27698	+	7.40794i	−0.307863	+	0.533235i	0.670031	+	0.742333i	$0.266281\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.2539			0.873056			0.436528	−	0.899691i	$-0.356208\pi$
$197$	12.2539			0.873056			0.436528	+	0.899691i	$0.356208\pi$
$198$		0			0
$199$	−1.71738			−0.121742			−0.0608710	−	0.998146i	$-0.519388\pi$
$199$	−1.71738			−0.121742			−0.0608710	+	0.998146i	$0.519388\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.37368	−	7.57544i	0.306972	−	0.531692i
$204$		0			0
$205$	−3.32056	−	5.75139i	−0.231918	−	0.401694i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.9439	−	18.9554i	−0.757006	−	1.31117i
$210$		0			0
$211$	1.26661	−	2.19384i	0.0871972	−	0.151030i	−0.819128	−	0.573611i	$-0.805542\pi$
$211$	1.26661	−	2.19384i	0.0871972	−	0.151030i	0.906325	+	0.422581i	$0.138876\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.55395			−0.446976
$216$		0			0
$217$	−15.8536			−1.07621
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.69832	−	8.13772i	0.316043	−	0.547403i
$222$		0			0
$223$	11.1659	+	19.3399i	0.747725	+	1.29510i	0.948911	+	0.315545i	$0.102187\pi$
$223$	11.1659	+	19.3399i	0.747725	+	1.29510i	−0.201185	+	0.979553i	$0.564479\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.20886	+	9.02201i	0.345724	+	0.598812i	0.985485	−	0.169762i	$-0.0542999\pi$
$227$	5.20886	+	9.02201i	0.345724	+	0.598812i	−0.639761	+	0.768574i	$0.720967\pi$
$228$		0			0
$229$	8.78169	−	15.2103i	0.580311	−	1.00513i	−0.415132	−	0.909761i	$-0.636264\pi$
$229$	8.78169	−	15.2103i	0.580311	−	1.00513i	0.995442	−	0.0953662i	$-0.0304022\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.2328			−1.25999			−0.629993	−	0.776601i	$-0.716942\pi$
$233$	−19.2328			−1.25999			−0.629993	+	0.776601i	$0.716942\pi$
$234$		0			0
$235$	−10.0398			−0.654924
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.25227	−	9.09720i	0.339741	−	0.588449i	−0.644643	−	0.764484i	$-0.722994\pi$
$239$	5.25227	−	9.09720i	0.339741	−	0.588449i	0.984384	+	0.176035i	$0.0563272\pi$
$240$		0			0
$241$	−4.66037	−	8.07200i	−0.300201	−	0.519963i	0.675980	−	0.736920i	$-0.263720\pi$
$241$	−4.66037	−	8.07200i	−0.300201	−	0.519963i	−0.976181	+	0.216956i	$0.930387\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.95623	+	12.0485i	0.444417	+	0.769753i
$246$		0			0
$247$	−8.76846	+	15.1874i	−0.557924	+	0.966352i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.9125			1.25686			0.628432	−	0.777865i	$-0.283697\pi$
$251$	19.9125			1.25686			0.628432	+	0.777865i	$0.283697\pi$
$252$		0			0
$253$	11.8155			0.742833
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.70868	+	9.88772i	−0.356098	+	0.616779i	−0.987305	−	0.158835i	$-0.949226\pi$
$257$	−5.70868	+	9.88772i	−0.356098	+	0.616779i	0.631207	+	0.775614i	$0.282560\pi$
$258$		0			0
$259$	−12.4998	−	21.6503i	−0.776701	−	1.34529i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.28568	−	3.95891i	−0.140941	−	0.244117i	0.786910	−	0.617067i	$-0.211679\pi$
$263$	−2.28568	−	3.95891i	−0.140941	−	0.244117i	−0.927851	+	0.372951i	$0.878346\pi$
$264$		0			0
$265$	−2.01906	+	3.49712i	−0.124030	+	0.214826i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.5954			−1.07281			−0.536405	−	0.843961i	$-0.680218\pi$
$269$	−17.5954			−1.07281			−0.536405	+	0.843961i	$0.680218\pi$
$270$		0			0
$271$	9.64113			0.585657			0.292828	−	0.956165i	$-0.405404\pi$
$271$	9.64113			0.585657			0.292828	+	0.956165i	$0.405404\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.29604	−	3.97686i	0.138456	−	0.239813i
$276$		0			0
$277$	−7.05211	−	12.2146i	−0.423720	−	0.733905i	0.572580	−	0.819849i	$-0.305943\pi$
$277$	−7.05211	−	12.2146i	−0.423720	−	0.733905i	−0.996300	+	0.0859443i	$0.972609\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.72793	+	2.99285i	0.103079	+	0.178539i	0.912952	−	0.408067i	$-0.133797\pi$
$281$	1.72793	+	2.99285i	0.103079	+	0.178539i	−0.809873	+	0.586606i	$0.800464\pi$
$282$		0			0
$283$	12.0911	−	20.9423i	0.718739	−	1.24489i	−0.242760	−	0.970086i	$-0.578053\pi$
$283$	12.0911	−	20.9423i	0.718739	−	1.24489i	0.961500	−	0.274807i	$-0.0886138\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.3700			−1.79268
$288$		0			0
$289$	−10.4773			−0.616314
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.62715	+	4.55035i	−0.153480	+	0.265834i	−0.932504	−	0.361159i	$-0.882381\pi$
$293$	−2.62715	+	4.55035i	−0.153480	+	0.265834i	0.779025	+	0.626993i	$0.215715\pi$
$294$		0			0
$295$	6.13567	+	10.6273i	0.357232	+	0.618744i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.73339	−	8.19847i	−0.273739	−	0.474130i
$300$		0			0
$301$	−14.9857	+	25.9559i	−0.863759	+	1.49607i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.59208			−0.549241
$306$		0			0
$307$	−1.31148			−0.0748503			−0.0374252	−	0.999299i	$-0.511916\pi$
$307$	−1.31148			−0.0748503			−0.0374252	+	0.999299i	$0.511916\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.43170	+	7.67594i	−0.251299	+	0.435263i	−0.963884	−	0.266324i	$-0.914191\pi$
$311$	−4.43170	+	7.67594i	−0.251299	+	0.435263i	0.712585	+	0.701586i	$0.247524\pi$
$312$		0			0
$313$	−1.12169	−	1.94282i	−0.0634014	−	0.109814i	0.832582	−	0.553901i	$-0.186862\pi$
$313$	−1.12169	−	1.94282i	−0.0634014	−	0.109814i	−0.895984	+	0.444087i	$0.853528\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.2873	+	17.8182i	0.577794	+	1.00077i	0.995732	+	0.0922935i	$0.0294198\pi$
$317$	10.2873	+	17.8182i	0.577794	+	1.00077i	−0.417937	+	0.908476i	$0.637247\pi$
$318$		0			0
$319$	−4.39192	+	7.60702i	−0.245900	+	0.425911i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.1732			−0.677337
$324$		0			0
$325$	−3.67925			−0.204088
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−22.9560	+	39.7610i	−1.26561	+	2.19210i
$330$		0			0
$331$	10.8938	+	18.8685i	0.598775	+	1.03711i	0.993002	+	0.118096i	$0.0376790\pi$
$331$	10.8938	+	18.8685i	0.598775	+	1.03711i	−0.394227	+	0.919013i	$0.628988\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.66972	+	9.82025i	0.309770	+	0.536537i
$336$		0			0
$337$	−6.22284	+	10.7783i	−0.338980	+	0.587130i	−0.984241	−	0.176832i	$-0.943415\pi$
$337$	−6.22284	+	10.7783i	−0.338980	+	0.587130i	0.645261	+	0.763962i	$0.276749\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	15.9197			0.862100
$342$		0			0
$343$	31.6108			1.70682
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.59736	+	14.8911i	−0.461530	+	0.799394i	−0.999037	−	0.0438652i	$-0.986033\pi$
$347$	−8.59736	+	14.8911i	−0.461530	+	0.799394i	0.537507	+	0.843259i	$0.319366\pi$
$348$		0			0
$349$	−4.74191	−	8.21322i	−0.253828	−	0.439644i	0.710748	−	0.703446i	$-0.248357\pi$
$349$	−4.74191	−	8.21322i	−0.253828	−	0.439644i	−0.964577	+	0.263803i	$0.915023\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.70868	+	11.6198i	0.357067	+	0.618458i	0.987469	−	0.157810i	$-0.0504434\pi$
$353$	6.70868	+	11.6198i	0.357067	+	0.618458i	−0.630402	+	0.776268i	$0.717110\pi$
$354$		0			0
$355$	−0.839627	+	1.45428i	−0.0445628	+	0.0771850i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.3970			0.654289			0.327144	−	0.944974i	$-0.393914\pi$
$359$	12.3970			0.654289			0.327144	+	0.944974i	$0.393914\pi$
$360$		0			0
$361$	3.71887			0.195730
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.56265	−	2.70659i	0.0817929	−	0.141670i
$366$		0			0
$367$	−10.2857	−	17.8153i	−0.536908	−	0.929952i	−0.999068	−	0.0431555i	$-0.986259\pi$
$367$	−10.2857	−	17.8153i	−0.536908	−	0.929952i	0.462160	−	0.886796i	$-0.347074\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.23321	+	15.9924i	0.479364	+	0.830283i
$372$		0			0
$373$	2.23154	−	3.86515i	0.115545	−	0.200130i	−0.802453	−	0.596716i	$-0.796472\pi$
$373$	2.23154	−	3.86515i	0.115545	−	0.200130i	0.917997	+	0.396586i	$0.129805\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.03776			0.362463
$378$		0			0
$379$	8.77660			0.450823			0.225412	−	0.974264i	$-0.427627\pi$
$379$	8.77660			0.450823			0.225412	+	0.974264i	$0.427627\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.60606	−	2.78177i	0.0820658	−	0.142142i	−0.822071	−	0.569384i	$-0.807182\pi$
$383$	1.60606	−	2.78177i	0.0820658	−	0.142142i	0.904137	+	0.427242i	$0.140515\pi$
$384$		0			0
$385$	−10.4998	−	18.1862i	−0.535120	−	0.926856i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.7957	+	22.1628i	0.648766	+	1.12370i	0.983418	+	0.181354i	$0.0580481\pi$
$389$	12.7957	+	22.1628i	0.648766	+	1.12370i	−0.334651	+	0.942342i	$0.608619\pi$
$390$		0			0
$391$	3.28568	−	5.69096i	0.166164	−	0.287804i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.2823			−0.668303
$396$		0			0
$397$	14.3902			0.722221			0.361111	−	0.932523i	$-0.382398\pi$
$397$	14.3902			0.722221			0.361111	+	0.932523i	$0.382398\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.0692	−	22.6365i	0.652645	−	1.13042i	−0.329833	−	0.944039i	$-0.606992\pi$
$401$	13.0692	−	22.6365i	0.652645	−	1.13042i	0.982479	−	0.186376i	$-0.0596742\pi$
$402$		0			0
$403$	−6.37757	−	11.0463i	−0.317689	−	0.550254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.5519	+	21.7406i	0.622176	+	1.07764i
$408$		0			0
$409$	−3.29096	+	5.70010i	−0.162727	+	0.281852i	−0.935846	−	0.352410i	$-0.885362\pi$
$409$	−3.29096	+	5.70010i	−0.162727	+	0.281852i	0.773119	+	0.634262i	$0.218696\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	56.1170			2.76134
$414$		0			0
$415$	−10.7855			−0.529439
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.37451	+	11.0410i	−0.311415	+	0.539387i	−0.978669	−	0.205443i	$-0.934136\pi$
$419$	−6.37451	+	11.0410i	−0.311415	+	0.539387i	0.667254	+	0.744831i	$0.267470\pi$
$420$		0			0
$421$	0.641128	+	1.11047i	0.0312467	+	0.0541208i	0.881226	−	0.472695i	$-0.156719\pi$
$421$	0.641128	+	1.11047i	0.0312467	+	0.0541208i	−0.849979	+	0.526816i	$0.823386\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.27698	−	2.21179i	−0.0619424	−	0.107287i
$426$		0			0
$427$	−21.9324	+	37.9880i	−1.06138	+	1.83837i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.89137			0.331946			0.165973	−	0.986130i	$-0.446924\pi$
$431$	6.89137			0.331946			0.165973	+	0.986130i	$0.446924\pi$
$432$		0			0
$433$	30.4551			1.46358			0.731790	−	0.681530i	$-0.238685\pi$
$433$	30.4551			1.46358			0.731790	+	0.681530i	$0.238685\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.13205	+	10.6210i	−0.293336	+	0.508072i
$438$		0			0
$439$	−14.8654	−	25.7477i	−0.709488	−	1.22887i	−0.965047	−	0.262077i	$-0.915593\pi$
$439$	−14.8654	−	25.7477i	−0.709488	−	1.22887i	0.255559	−	0.966794i	$-0.417741\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.16482	+	5.48163i	0.150365	+	0.260440i	0.931362	−	0.364095i	$-0.118622\pi$
$443$	3.16482	+	5.48163i	0.150365	+	0.260440i	−0.780997	+	0.624535i	$0.785288\pi$
$444$		0			0
$445$	−2.02453	+	3.50658i	−0.0959717	+	0.166228i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−25.8456			−1.21973			−0.609866	−	0.792505i	$-0.708777\pi$
$449$	−25.8456			−1.21973			−0.609866	+	0.792505i	$0.708777\pi$
$450$		0			0
$451$	30.4966			1.43603
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.41264	+	14.5711i	−0.394391	+	0.683105i
$456$		0			0
$457$	17.6545	+	30.5786i	0.825845	+	1.43041i	0.901272	+	0.433254i	$0.142635\pi$
$457$	17.6545	+	30.5786i	0.825845	+	1.43041i	−0.0754270	+	0.997151i	$0.524032\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.55941	+	2.70098i	0.0726291	+	0.125797i	0.900053	−	0.435781i	$-0.143528\pi$
$461$	1.55941	+	2.70098i	0.0726291	+	0.125797i	−0.827424	+	0.561578i	$0.810194\pi$
$462$		0			0
$463$	−1.58736	+	2.74939i	−0.0737708	+	0.127775i	−0.900551	−	0.434750i	$-0.856837\pi$
$463$	−1.58736	+	2.74939i	−0.0737708	+	0.127775i	0.826780	+	0.562525i	$0.190170\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.2808			0.938482			0.469241	−	0.883070i	$-0.344528\pi$
$467$	20.2808			0.938482			0.469241	+	0.883070i	$0.344528\pi$
$468$		0			0
$469$	51.8554			2.39446
$470$		0			0
$471$		0			0
$472$		0			0
$473$	15.0481	−	26.0641i	0.691914	−	1.19843i
$474$		0			0
$475$	2.38322	+	4.12785i	0.109349	+	0.189399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.03813	+	8.72629i	0.230198	+	0.398714i	0.957866	−	0.287215i	$-0.0927293\pi$
$479$	5.03813	+	8.72629i	0.230198	+	0.398714i	−0.727668	+	0.685929i	$0.759396\pi$
$480$		0			0
$481$	10.0568	−	17.4189i	0.458552	−	0.794235i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.3792			−0.789150
$486$		0			0
$487$	28.3589			1.28506			0.642532	−	0.766259i	$-0.277884\pi$
$487$	28.3589			1.28506			0.642532	+	0.766259i	$0.277884\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16.4560	+	28.5027i	−0.742651	+	1.28631i	0.208633	+	0.977994i	$0.433099\pi$
$491$	−16.4560	+	28.5027i	−0.742651	+	1.28631i	−0.951284	+	0.308315i	$0.900235\pi$
$492$		0			0
$493$	2.44263	+	4.23076i	0.110010	+	0.190544i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.83963	+	6.65043i	0.172231	+	0.298313i
$498$		0			0
$499$	2.50546	−	4.33959i	0.112160	−	0.194267i	−0.804481	−	0.593978i	$-0.797556\pi$
$499$	2.50546	−	4.33959i	0.112160	−	0.194267i	0.916641	+	0.399712i	$0.130890\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	42.1916			1.88123			0.940615	−	0.339477i	$-0.110250\pi$
$503$	42.1916			1.88123			0.940615	+	0.339477i	$0.110250\pi$
$504$		0			0
$505$	13.1460			0.584991
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.61660	+	14.9244i	−0.381924	+	0.661512i	−0.991337	−	0.131341i	$-0.958072\pi$
$509$	−8.61660	+	14.9244i	−0.381924	+	0.661512i	0.609413	+	0.792853i	$0.291405\pi$
$510$		0			0
$511$	−7.14603	−	12.3773i	−0.316122	−	0.547539i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.75245	+	9.96354i	0.253483	+	0.439046i
$516$		0			0
$517$	23.0517	−	39.9268i	1.01381	−	1.75598i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.42901			0.237849			0.118925	−	0.992903i	$-0.462055\pi$
$521$	5.42901			0.237849			0.118925	+	0.992903i	$0.462055\pi$
$522$		0			0
$523$	−6.68128			−0.292152			−0.146076	−	0.989273i	$-0.546664\pi$
$523$	−6.68128			−0.292152			−0.146076	+	0.989273i	$0.546664\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.42699	−	7.66776i	0.192843	−	0.334013i
$528$		0			0
$529$	8.18980	+	14.1851i	0.356078	+	0.616746i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.2172	−	21.1608i	−0.529186	−	0.916576i
$534$		0			0
$535$	2.76198	−	4.78389i	0.119411	−	0.206826i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−63.8871			−2.75181
$540$		0			0
$541$	35.3087			1.51804			0.759020	−	0.651067i	$-0.225678\pi$
$541$	35.3087			1.51804			0.759020	+	0.651067i	$0.225678\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.36905	+	14.4956i	−0.358491	+	0.620924i
$546$		0			0
$547$	5.78105	+	10.0131i	0.247180	+	0.428128i	0.962742	−	0.270421i	$-0.0871629\pi$
$547$	5.78105	+	10.0131i	0.247180	+	0.428128i	−0.715563	+	0.698549i	$0.753830\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.55867	−	7.89585i	−0.194206	−	0.336374i
$552$		0			0
$553$	−30.3700	+	52.6023i	−1.29146	+	2.23688i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.0415			0.764441			0.382220	−	0.924071i	$-0.375160\pi$
$557$	18.0415			0.764441			0.382220	+	0.924071i	$0.375160\pi$
$558$		0			0
$559$	−24.1137			−1.01990
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.28086	+	2.21852i	−0.0539820	+	0.0934995i	−0.891754	−	0.452521i	$-0.850525\pi$
$563$	−1.28086	+	2.21852i	−0.0539820	+	0.0934995i	0.837772	+	0.546021i	$0.183858\pi$
$564$		0			0
$565$	−1.89376	−	3.28009i	−0.0796711	−	0.137994i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.3502	−	17.9270i	−0.433902	−	0.751540i	0.563304	−	0.826250i	$-0.309530\pi$
$569$	−10.3502	−	17.9270i	−0.433902	−	0.751540i	−0.997205	+	0.0747101i	$0.976197\pi$
$570$		0			0
$571$	−3.58338	+	6.20659i	−0.149960	+	0.259738i	−0.931212	−	0.364477i	$-0.881248\pi$
$571$	−3.58338	+	6.20659i	−0.149960	+	0.259738i	0.781253	+	0.624215i	$0.214581\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.57301			−0.107302
$576$		0			0
$577$	−27.2463			−1.13428			−0.567140	−	0.823622i	$-0.691950\pi$
$577$	−27.2463			−1.13428			−0.567140	+	0.823622i	$0.691950\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.6611	+	42.7143i	−1.02312	+	1.77209i
$582$		0			0
$583$	−9.27170	−	16.0590i	−0.383994	−	0.665098i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.24746	−	14.2850i	−0.340409	−	0.589606i	0.644100	−	0.764942i	$-0.277232\pi$
$587$	−8.24746	−	14.2850i	−0.340409	−	0.589606i	−0.984509	+	0.175336i	$0.943899\pi$
$588$		0			0
$589$	−8.26207	+	14.3103i	−0.340433	+	0.589647i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.2819			0.586487			0.293244	−	0.956038i	$-0.405265\pi$
$593$	14.2819			0.586487			0.293244	+	0.956038i	$0.405265\pi$
$594$		0			0
$595$	−11.6793			−0.478803
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.1951	+	19.3904i	−0.457419	+	0.792272i	−0.998824	−	0.0484897i	$-0.984559\pi$
$599$	−11.1951	+	19.3904i	−0.457419	+	0.792272i	0.541405	+	0.840762i	$0.317893\pi$
$600$		0			0
$601$	6.05015	+	10.4792i	0.246791	+	0.427454i	0.962634	−	0.270807i	$-0.0872906\pi$
$601$	6.05015	+	10.4792i	0.246791	+	0.427454i	−0.715843	+	0.698262i	$0.753957\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.04359	+	8.73575i	0.205051	+	0.355159i
$606$		0			0
$607$	10.1659	−	17.6079i	0.412622	−	0.714682i	−0.582554	−	0.812792i	$-0.697946\pi$
$607$	10.1659	−	17.6079i	0.412622	−	0.714682i	0.995176	+	0.0981100i	$0.0312797\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−36.9389			−1.49439
$612$		0			0
$613$	24.8853			1.00511			0.502553	−	0.864546i	$-0.332394\pi$
$613$	24.8853			1.00511			0.502553	+	0.864546i	$0.332394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.6994	−	39.3165i	0.913844	−	1.58282i	0.105259	−	0.994445i	$-0.466433\pi$
$617$	22.6994	−	39.3165i	0.913844	−	1.58282i	0.808585	−	0.588380i	$-0.200234\pi$
$618$		0			0
$619$	9.71376	+	16.8247i	0.390429	+	0.676243i	0.992506	−	0.122195i	$-0.0389933\pi$
$619$	9.71376	+	16.8247i	0.390429	+	0.676243i	−0.602077	+	0.798438i	$0.705660\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.25818	+	16.0356i	0.370921	+	0.642454i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13.9619			0.556696
$630$		0			0
$631$	−7.78752			−0.310016			−0.155008	−	0.987913i	$-0.549540\pi$
$631$	−7.78752			−0.310016			−0.155008	+	0.987913i	$0.549540\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.57218	−	9.65130i	0.221125	−	0.383000i
$636$		0			0
$637$	25.5937	+	44.3297i	1.01406	+	1.75640i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.6981	−	25.4579i	−0.580541	−	1.00553i	−0.995415	−	0.0956483i	$-0.969508\pi$
$641$	−14.6981	−	25.4579i	−0.580541	−	1.00553i	0.414874	−	0.909879i	$-0.363826\pi$
$642$		0			0
$643$	4.65066	−	8.05518i	0.183404	−	0.317665i	−0.759633	−	0.650351i	$-0.774622\pi$
$643$	4.65066	−	8.05518i	0.183404	−	0.317665i	0.943038	+	0.332686i	$0.107955\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−40.4647			−1.59083			−0.795417	−	0.606063i	$-0.792748\pi$
$647$	−40.4647			−1.59083			−0.795417	+	0.606063i	$0.792748\pi$
$648$		0			0
$649$	−56.3509			−2.21197
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.32546	+	12.6881i	−0.286668	+	0.496523i	−0.973012	−	0.230754i	$-0.925881\pi$
$653$	−7.32546	+	12.6881i	−0.286668	+	0.496523i	0.686345	+	0.727276i	$0.259214\pi$
$654$		0			0
$655$	−10.9125	−	18.9009i	−0.426385	−	0.738521i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.51380	+	4.35403i	0.0979237	+	0.169609i	0.910825	−	0.412793i	$-0.135447\pi$
$659$	2.51380	+	4.35403i	0.0979237	+	0.169609i	−0.812901	+	0.582401i	$0.802113\pi$
$660$		0			0
$661$	−15.9128	+	27.5618i	−0.618937	+	1.07203i	0.370743	+	0.928736i	$0.379103\pi$
$661$	−15.9128	+	27.5618i	−0.618937	+	1.07203i	−0.989680	+	0.143295i	$0.954230\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	21.7970			0.845250
$666$		0			0
$667$	4.92172			0.190570
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.0238	−	38.1463i	0.850219	−	1.47262i
$672$		0			0
$673$	−5.45077	−	9.44101i	−0.210112	−	0.363924i	0.741638	−	0.670801i	$-0.234049\pi$
$673$	−5.45077	−	9.44101i	−0.210112	−	0.363924i	−0.951749	+	0.306877i	$0.900716\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.0047	−	24.2569i	−0.538245	−	0.932268i	−0.998999	−	0.0447397i	$-0.985754\pi$
$677$	−14.0047	−	24.2569i	−0.538245	−	0.932268i	0.460754	−	0.887528i	$-0.347579\pi$
$678$		0			0
$679$	−39.7377	+	68.8278i	−1.52499	+	2.64137i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.19544			−0.275326			−0.137663	−	0.990479i	$-0.543959\pi$
$683$	−7.19544			−0.275326			−0.137663	+	0.990479i	$0.543959\pi$
$684$		0			0
$685$	−4.44569			−0.169861
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.42865	+	12.8668i	−0.283009	+	0.490186i
$690$		0			0
$691$	11.2713	+	19.5225i	0.428782	+	0.742672i	0.996765	−	0.0803682i	$-0.0256096\pi$
$691$	11.2713	+	19.5225i	0.428782	+	0.742672i	−0.567984	+	0.823040i	$0.692276\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.58338	−	9.67069i	−0.211790	−	0.366830i
$696$		0			0
$697$	8.48056	−	14.6888i	0.321224	−	0.556376i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.7098			0.744428			0.372214	−	0.928147i	$-0.378599\pi$
$701$	19.7098			0.744428			0.372214	+	0.928147i	$0.378599\pi$
$702$		0			0
$703$	−26.0570			−0.982758
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.0585	−	52.0628i	1.13047	−	1.95802i
$708$		0			0
$709$	−12.0643	−	20.8960i	−0.453085	−	0.784766i	0.545491	−	0.838117i	$-0.316343\pi$
$709$	−12.0643	−	20.8960i	−0.453085	−	0.784766i	−0.998576	+	0.0533508i	$0.983010\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.46003	−	7.72500i	−0.167029	−	0.289303i
$714$		0			0
$715$	8.44771	−	14.6319i	0.315926	−	0.547201i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.79732			0.328085			0.164042	−	0.986453i	$-0.447547\pi$
$719$	8.79732			0.328085			0.164042	+	0.986453i	$0.447547\pi$
$720$		0			0
$721$	52.6121			1.95938
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.956412	−	1.65655i	0.0355202	−	0.0615229i
$726$		0			0
$727$	−2.62123	−	4.54011i	−0.0972162	−	0.168383i	0.813315	−	0.581823i	$-0.197660\pi$
$727$	−2.62123	−	4.54011i	−0.0972162	−	0.168383i	−0.910531	+	0.413440i	$0.864327\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.36923	−	14.4959i	−0.309547	−	0.536152i
$732$		0			0
$733$	17.0589	−	29.5468i	0.630083	−	1.09134i	−0.357451	−	0.933932i	$-0.616354\pi$
$733$	17.0589	−	29.5468i	0.630083	−	1.09134i	0.987534	−	0.157404i	$-0.0503125\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−52.0716			−1.91808
$738$		0			0
$739$	5.53174			0.203488			0.101744	−	0.994811i	$-0.467558\pi$
$739$	5.53174			0.203488			0.101744	+	0.994811i	$0.467558\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.77234	−	13.4621i	0.285140	−	0.493876i	−0.687503	−	0.726181i	$-0.741293\pi$
$743$	7.77234	−	13.4621i	0.285140	−	0.493876i	0.972643	+	0.232305i	$0.0746267\pi$
$744$		0			0
$745$	6.54341	+	11.3335i	0.239732	+	0.415228i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.6306	−	21.8768i	−0.461511	−	0.799361i
$750$		0			0
$751$	11.2472	−	19.4807i	0.410416	−	0.710861i	−0.584519	−	0.811380i	$-0.698717\pi$
$751$	11.2472	−	19.4807i	0.410416	−	0.710861i	0.994935	+	0.100519i	$0.0320503\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.358872			−0.0130607
$756$		0			0
$757$	−14.5714			−0.529605			−0.264802	−	0.964303i	$-0.585307\pi$
$757$	−14.5714			−0.529605			−0.264802	+	0.964303i	$0.585307\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.90736	+	11.9639i	−0.250392	+	0.433691i	−0.963634	−	0.267227i	$-0.913893\pi$
$761$	−6.90736	+	11.9639i	−0.250392	+	0.433691i	0.713242	+	0.700918i	$0.247226\pi$
$762$		0			0
$763$	38.2718	+	66.2887i	1.38553	+	2.39981i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.5747	+	39.1005i	0.815124	+	1.41184i
$768$		0			0
$769$	0.0643136	−	0.111394i	0.00231921	−	0.00401698i	−0.864863	−	0.502007i	$-0.832595\pi$
$769$	0.0643136	−	0.111394i	0.00231921	−	0.00401698i	0.867183	+	0.497990i	$0.165928\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	44.5427			1.60209			0.801044	−	0.598605i	$-0.204278\pi$
$773$	44.5427			1.60209			0.801044	+	0.598605i	$0.204278\pi$
$774$		0			0
$775$	−3.46677			−0.124530
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.8272	+	27.4136i	−0.567070	+	0.982194i
$780$		0			0
$781$	−3.85563	−	6.67815i	−0.137965	−	0.238963i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.53489	−	11.3188i	−0.233240	−	0.403984i
$786$		0			0
$787$	16.3932	−	28.3939i	0.584355	−	1.01213i	−0.410601	−	0.911815i	$-0.634681\pi$
$787$	16.3932	−	28.3939i	0.584355	−	1.01213i	0.994956	−	0.100317i	$-0.0319858\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.3204			−0.615842
$792$		0			0
$793$	−35.2917			−1.25324
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26.9383	−	46.6584i	0.954202	−	1.65273i	0.218019	−	0.975944i	$-0.430040\pi$
$797$	26.9383	−	46.6584i	0.954202	−	1.65273i	0.736183	−	0.676783i	$-0.236626\pi$
$798$		0			0
$799$	−12.8206	−	22.2059i	−0.453559	−	0.785587i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.17582	+	12.4289i	0.253229	+	0.438606i
$804$		0			0
$805$	−5.88322	+	10.1900i	−0.207356	+	0.359151i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	44.3890			1.56064			0.780318	−	0.625383i	$-0.215057\pi$
$809$	44.3890			1.56064			0.780318	+	0.625383i	$0.215057\pi$
$810$		0			0
$811$	−23.6339			−0.829898			−0.414949	−	0.909845i	$-0.636201\pi$
$811$	−23.6339			−0.829898			−0.414949	+	0.909845i	$0.636201\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.39358	−	16.2702i	0.329043	−	0.569919i
$816$		0			0
$817$	15.6195	+	27.0537i	0.546456	+	0.946490i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.40921	−	5.90492i	−0.118982	−	0.206083i	0.800382	−	0.599490i	$-0.204630\pi$
$821$	−3.40921	−	5.90492i	−0.118982	−	0.206083i	−0.919365	+	0.393407i	$0.871296\pi$
$822$		0			0
$823$	−21.1656	+	36.6598i	−0.737785	+	1.27788i	0.215706	+	0.976458i	$0.430795\pi$
$823$	−21.1656	+	36.6598i	−0.737785	+	1.27788i	−0.953491	+	0.301423i	$0.902538\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.3700			−1.05607			−0.528034	−	0.849223i	$-0.677071\pi$
$827$	−30.3700			−1.05607			−0.528034	+	0.849223i	$0.677071\pi$
$828$		0			0
$829$	−45.8420			−1.59216			−0.796078	−	0.605193i	$-0.793096\pi$
$829$	−45.8420			−1.59216			−0.796078	+	0.605193i	$0.793096\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17.7659	+	30.7714i	−0.615551	+	1.06617i
$834$		0			0
$835$	7.12613	+	12.3428i	0.246610	+	0.427141i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.62512	+	9.74300i	0.194201	+	0.336366i	0.946638	−	0.322298i	$-0.104455\pi$
$839$	5.62512	+	9.74300i	0.194201	+	0.336366i	−0.752437	+	0.658664i	$0.771122\pi$
$840$		0			0
$841$	12.6706	−	21.9460i	0.436916	−	0.756760i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.536913			−0.0184704
$846$		0			0
$847$	46.1288			1.58500
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.03304	−	12.1816i	0.241090	−	0.417580i
$852$		0			0
$853$	−2.67417	−	4.63180i	−0.0915619	−	0.158590i	0.816607	−	0.577195i	$-0.195853\pi$
$853$	−2.67417	−	4.63180i	−0.0915619	−	0.158590i	−0.908169	+	0.418605i	$0.862519\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.9646	+	44.9719i	0.886933	+	1.53621i	0.843482	+	0.537157i	$0.180502\pi$
$857$	25.9646	+	44.9719i	0.886933	+	1.53621i	0.0434504	+	0.999056i	$0.486165\pi$
$858$		0			0
$859$	14.7117	−	25.4815i	0.501958	−	0.869417i	−0.498039	−	0.867154i	$-0.665947\pi$
$859$	14.7117	−	25.4815i	0.501958	−	0.869417i	0.999997	−	0.00226239i	$-0.000720142\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.9635			0.543405			0.271703	−	0.962381i	$-0.412413\pi$
$863$	15.9635			0.543405			0.271703	+	0.962381i	$0.412413\pi$
$864$		0			0
$865$	−9.18416			−0.312271
$866$		0			0
$867$		0			0
$868$		0			0
$869$	30.4966	−	52.8216i	1.03453	−	1.79185i
$870$		0			0
$871$	20.8604	+	36.1312i	0.706826	+	1.22426i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.28651	+	3.96035i	0.0772980	+	0.133884i
$876$		0			0
$877$	−2.71432	+	4.70135i	−0.0916562	+	0.158753i	−0.908208	−	0.418519i	$-0.862549\pi$
$877$	−2.71432	+	4.70135i	−0.0916562	+	0.158753i	0.816552	+	0.577272i	$0.195883\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.2676			−0.345926			−0.172963	−	0.984928i	$-0.555334\pi$
$881$	−10.2676			−0.345926			−0.172963	+	0.984928i	$0.555334\pi$
$882$		0			0
$883$	−12.6184			−0.424642			−0.212321	−	0.977200i	$-0.568102\pi$
$883$	−12.6184			−0.424642			−0.212321	+	0.977200i	$0.568102\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.8015	−	32.5652i	0.631293	−	1.09343i	−0.355995	−	0.934488i	$-0.615858\pi$
$887$	18.8015	−	32.5652i	0.631293	−	1.09343i	0.987288	−	0.158943i	$-0.0508086\pi$
$888$		0			0
$889$	−25.4817	−	44.1356i	−0.854628	−	1.48026i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.9270	+	41.4427i	0.800686	+	1.38683i
$894$		0			0
$895$	9.39358	−	16.2702i	0.313993	−	0.543851i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.63133			0.221167
$900$		0			0
$901$	−10.3132			−0.343582
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.95641	−	10.3168i	0.197998	−	0.342942i
$906$		0			0
$907$	−14.1556	−	24.5181i	−0.470028	−	0.814112i	0.529385	−	0.848382i	$-0.322423\pi$
$907$	−14.1556	−	24.5181i	−0.470028	−	0.814112i	−0.999413	+	0.0342699i	$0.989089\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.80658	+	6.59320i	0.126118	+	0.218442i	0.922169	−	0.386786i	$-0.126415\pi$
$911$	3.80658	+	6.59320i	0.126118	+	0.218442i	−0.796052	+	0.605229i	$0.793082\pi$
$912$		0			0
$913$	24.7639	−	42.8924i	0.819566	−	1.41953i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−99.8057			−3.29587
$918$		0			0
$919$	41.2182			1.35966			0.679832	−	0.733368i	$-0.262053\pi$
$919$	41.2182			1.35966			0.679832	+	0.733368i	$0.262053\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.08920	+	5.35065i	−0.101682	+	0.176119i
$924$		0			0
$925$	−2.73339	−	4.73437i	−0.0898732	−	0.155665i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.6128	+	30.5063i	0.577857	+	1.00088i	0.995725	+	0.0923702i	$0.0294443\pi$
$929$	17.6128	+	30.5063i	0.577857	+	1.00088i	−0.417867	+	0.908508i	$0.637222\pi$
$930$		0			0
$931$	33.1564	−	57.4286i	1.08666	−	1.88214i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.7279			0.383545
$936$		0			0
$937$	−15.7113			−0.513265			−0.256632	−	0.966509i	$-0.582613\pi$
$937$	−15.7113			−0.513265			−0.256632	+	0.966509i	$0.582613\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20.2594	+	35.0903i	−0.660437	+	1.14391i	0.320064	+	0.947396i	$0.396296\pi$
$941$	−20.2594	+	35.0903i	−0.660437	+	1.14391i	−0.980501	+	0.196515i	$0.937038\pi$
$942$		0			0
$943$	−8.54386	−	14.7984i	−0.278226	−	0.481902i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.0266	−	22.5627i	−0.423307	−	0.733189i	0.572954	−	0.819588i	$-0.305797\pi$
$947$	−13.0266	−	22.5627i	−0.423307	−	0.733189i	−0.996261	+	0.0863989i	$0.972464\pi$
$948$		0			0
$949$	5.74939	−	9.95824i	0.186633	−	0.323258i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.24264			−0.267005			−0.133503	−	0.991048i	$-0.542622\pi$
$953$	−8.24264			−0.267005			−0.133503	+	0.991048i	$0.542622\pi$
$954$		0			0
$955$	−4.74571			−0.153567
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.1651	+	17.6065i	−0.328248	+	0.568542i
$960$		0			0
$961$	9.49074	+	16.4384i	0.306153	+	0.530272i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.27698	+	7.40794i	0.137681	+	0.238470i
$966$		0			0
$967$	6.25818	−	10.8395i	0.201250	−	0.348574i	−0.747682	−	0.664057i	$-0.768833\pi$
$967$	6.25818	−	10.8395i	0.201250	−	0.348574i	0.948931	+	0.315483i	$0.102166\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	34.5180			1.10774			0.553868	−	0.832604i	$-0.313151\pi$
$971$	34.5180			1.10774			0.553868	+	0.832604i	$0.313151\pi$
$972$		0			0
$973$	−51.0657			−1.63709
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26.3164	+	45.5813i	−0.841936	+	1.45828i	0.0463193	+	0.998927i	$0.485251\pi$
$977$	−26.3164	+	45.5813i	−0.841936	+	1.45828i	−0.888256	+	0.459350i	$0.848082\pi$
$978$		0			0
$979$	−9.29678	−	16.1025i	−0.297126	−	0.514638i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.9082	−	31.0179i	−0.571183	−	0.989318i	−0.996445	−	0.0842478i	$-0.973151\pi$
$983$	−17.9082	−	31.0179i	−0.571183	−	0.989318i	0.425262	−	0.905070i	$-0.360182\pi$
$984$		0			0
$985$	6.12697	−	10.6122i	0.195221	−	0.338133i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.8634			−0.536225
$990$		0			0
$991$	−42.7203			−1.35706			−0.678528	−	0.734574i	$-0.737382\pi$
$991$	−42.7203			−1.35706			−0.678528	+	0.734574i	$0.737382\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.858691	+	1.48730i	−0.0272223	+	0.0471504i
$996$		0			0
$997$	−23.4571	−	40.6290i	−0.742895	−	1.28673i	−0.951172	−	0.308662i	$-0.900119\pi$
$997$	−23.4571	−	40.6290i	−0.742895	−	1.28673i	0.208276	−	0.978070i	$-0.433215\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.q.l.1441.1		8
3.2	odd	2		720.2.q.l.481.3		8
4.3	odd	2		1080.2.q.e.361.4		8
9.2	odd	6		720.2.q.l.241.3		8
9.4	even	3		6480.2.a.bz.1.4		4
9.5	odd	6		6480.2.a.cb.1.4		4
9.7	even	3	inner	2160.2.q.l.721.1		8
12.11	even	2		360.2.q.e.121.2	✓	8
36.7	odd	6		1080.2.q.e.721.4		8
36.11	even	6		360.2.q.e.241.2	yes	8
36.23	even	6		3240.2.a.u.1.1		4
36.31	odd	6		3240.2.a.s.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.2	✓	8	12.11	even	2
360.2.q.e.241.2	yes	8	36.11	even	6
720.2.q.l.241.3		8	9.2	odd	6
720.2.q.l.481.3		8	3.2	odd	2
1080.2.q.e.361.4		8	4.3	odd	2
1080.2.q.e.721.4		8	36.7	odd	6
2160.2.q.l.721.1		8	9.7	even	3	inner
2160.2.q.l.1441.1		8	1.1	even	1	trivial
3240.2.a.s.1.1		4	36.31	odd	6
3240.2.a.u.1.1		4	36.23	even	6
6480.2.a.bz.1.4		4	9.4	even	3
6480.2.a.cb.1.4		4	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(-2.28651 - 3.96035i) q^{7} +(2.29604 + 3.97686i) q^{11} +(1.83963 - 3.18633i) q^{13} +2.55395 q^{17} -4.76643 q^{19} +(1.28651 - 2.22830i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(0.956412 + 1.65655i) q^{29} +(1.73339 - 3.00231i) q^{31} -4.57301 q^{35} +5.46677 q^{37} +(3.32056 - 5.75139i) q^{41} +(-3.27698 - 5.67589i) q^{43} +(-5.01989 - 8.69471i) q^{47} +(-6.95623 + 12.0485i) q^{49} -4.03813 q^{53} +4.59208 q^{55} +(-6.13567 + 10.6273i) q^{59} +(-4.79604 - 8.30698i) q^{61} +(-1.83963 - 3.18633i) q^{65} +(-5.66972 + 9.82025i) q^{67} -1.67925 q^{71} +3.12530 q^{73} +(10.4998 - 18.1862i) q^{77} +(-6.64113 - 11.5028i) q^{79} +(-5.39275 - 9.34051i) q^{83} +(1.27698 - 2.21179i) q^{85} -4.04905 q^{89} -16.8253 q^{91} +(-2.38322 + 4.12785i) q^{95} +(-8.68962 - 15.0509i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} - q^{7} - q^{11} - 4 q^{13} - 10 q^{17} - 2 q^{19} - 7 q^{23} - 4 q^{25} + 7 q^{29} - 2 q^{31} - 2 q^{35} + 12 q^{37} + 12 q^{41} - 11 q^{43} - 7 q^{47} - 3 q^{49} - 24 q^{53} - 2 q^{55}+ \cdots - q^{97}+O(q^{100}) \) 8 * q + 4 * q^5 - q^7 - q^11 - 4 * q^13 - 10 * q^17 - 2 * q^19 - 7 * q^23 - 4 * q^25 + 7 * q^29 - 2 * q^31 - 2 * q^35 + 12 * q^37 + 12 * q^41 - 11 * q^43 - 7 * q^47 - 3 * q^49 - 24 * q^53 - 2 * q^55 - 11 * q^59 - 19 * q^61 + 4 * q^65 - 10 * q^67 + 24 * q^71 + 18 * q^73 + 32 * q^77 - 24 * q^79 - 23 * q^83 - 5 * q^85 - 42 * q^89 - 28 * q^91 - q^95 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more